The build-up of the colour–magnitude relation as a function of environment

Mon. Not. R. Astron. Soc. 362, 268–288 (2005) doi:10.1111/j.1365-2966.2005.09300.x

The build-up of the colour–magnitude relation as a functionof environment

Masayuki Tanaka,1� Tadayuki Kodama,2,3 Nobuo Arimoto,2 Sadanori Okamura,1,4

Keiichi Umetsu,5 Kazuhiro Shimasaku,1,4 Ichi Tanaka6 and Toru Yamada2

1Department of Astronomy, School of Science, University of Tokyo, Tokyo 113–0033, Japan2National Astronomical Observatory of Japan, Mitaka, Tokyo 181–8588, Japan3European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748, Garching, Germany4Research Center for the Early Universe, School of Science, University of Tokyo, Tokyo 113–0033, Japan5Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 106, Taiwan6Astronomical Institute, Tohoku University, Aoba-ku, Sendai 980–8578, Japan

Accepted 2005 June 16. Received 2005 May 3; in original form 2005 January 26

ABSTRACTWe discuss the environmental dependence of galaxy evolution based on deep panoramic imag-ing of two distant clusters, RX J0152.7–1357 at z = 0.83 and CL0016+1609 at z = 0.55,taken with the Subaru Prime Focus Camera on the Subaru Telescope as part of the PanoramicImaging and Spectroscopy of Cluster Evolution with Subaru project. By combining with theSloan Digital Sky Survey data as a local counterpart for comparison, we construct a large sam-ple of galaxies that spans wide ranges in environment, time and stellar mass (or luminosity).This allows us to conduct systematic and statistical analyses of the photometric properties ofgalaxies based on the colour–density diagrams, colour–magnitude relations, and luminosityfunctions. We find that colours of galaxies, especially those of faint galaxies (MV > M

∗V + 1),

change from blue to red at a break density as we go to denser regions. This trend is observedat all redshifts in our sample. Based on local and global densities of galaxies, we classify threeenvironments – field, groups and clusters – and look into the environmental dependence ofgalaxies in detail. In particular, we quantify how the colour–magnitude relation is built up as afunction of environment. We show that the bright end of the cluster colour–magnitude relationis already built at z = 0.83, while the faint end is possibly still in the process of build-up. Incontrast to this, the bright end of the field colour–magnitude relation has been vigorously builtall the way down to the present-day and the build-up at the faint end has not started yet. Apossible interpretation of these results is that galaxies evolve in a ‘down-sizing’ fashion. Thatis, massive galaxies complete their star formation first and the truncation of star formation ispropagated to smaller objects as time progresses. This trend is likely to depend on environmentsince the build-up of the colour–magnitude relation is delayed in lower density environments.Therefore, we may suggest that the evolution of galaxies took place earliest in massive galaxiesand in high-density regions, and it is delayed in less massive galaxies and in lower densityregions. Further studies are, however, obviously needed to confirm the observed trends andestablish the ‘down-sizing’ picture.

Key words: galaxies: clusters: individual: CL0016+1609 – galaxies: clusters: individual:RX J0152.7–1357 – galaxies: evolution – galaxies: fundamental parameters – galaxies: lumi-nosity function, mass function.

�E-mail: [email protected]

1 I N T RO D U C T I O N

Intensive studies of galaxy properties, such as star formation ratesand morphology, have significantly improved our understanding ofgalaxies in the Universe. It is, however, still unclear how galaxies

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evolve over the Hubble time. This is due to the complex nature ofgalaxy properties; galaxy properties depend not only on time, butalso on environment and mass (luminosity). These three axes are re-lated to one another and they characterize galaxy evolution. Despitetheir obvious importance, however, galaxy properties along thesethree axes have not been viewed simultaneously and systematically,due to the limited previous data sets, none of which can cover all ofthese three axes. Based on the wide-field Subaru data and the largeamount of Sloan Digital Sky Survey (SDSS) data, we present in thispaper a comprehensive study of star formation activity of galaxiesalong the three axes.

It is now well known that galaxy properties depend on the environ-ment in which galaxies reside. This environmental dependence ofgalaxy properties was first quantitatively studied by Dressler (1980),who showed the morphology–density relation based on 55 nearbyclusters. There is a clear trend that early-type galaxies are pref-erentially located in high-density regions, while late-type galaxiestend to be located in lower density regions. Following this work,intensive studies of environmental dependence of galaxy propertieshave been carried out, and have strengthened or extended Dressler’sresult (e.g. Postman & Geller 1984; Whitmore, Gilmore & Jones1993; Balogh et al. 1997; Dressler et al. 1997; Balogh et al. 1998;Hashimoto et al. 1998; Lubin et al. 1998; Oke, Postman & Lubin1998; Postman, Lubin & Oke 1998; van Dokkum et al. 1998; Baloghet al. 1999; Poggianti et al. 1999; Lubin et al. 2000; Couch et al.2001; Kodama et al. 2001a; Postman, Lubin & Oke 2001; Lewiset al. 2002; Lubin, Oke & Postman 2002; Blanton et al. 2003a;Gomez et al. 2003; Goto et al. 2003a; Hogg et al. 2003; Treuet al. 2003; Balogh et al. 2004a,b; Smith et al. 2005; Tanaka et al.2004).

Galaxy properties are also known to depend on the mass (lu-minosity) of galaxies. Recently conducted large surveys, such as2dF (Colless et al. 2003) and SDSS (York et al. 2000), revealedthat galaxy properties show strong bimodality in their distribution(Strateva et al. 2001; Blanton et al. 2003a; Kauffmann et al. 2003;Balogh et al. 2004a; Kauffmann et al. 2004; Tanaka et al. 2004).That is, there are two distinct populations: red early-type galaxiesand blue late-type galaxies. This bimodality is found to be a strongfunction of the mass of galaxies in the sense that massive galaxiestend to be red early-type galaxies, while less massive galaxies tendto be blue late-type galaxies (Kauffmann et al. 2003; Balogh et al.2004a; Baldry et al. 2004; Bell et al. 2004; Kauffmann et al. 2004)

Galaxies evolve with time, of course, as a natural consequence ofstellar evolution. It is now 20 years since Butcher & Oemler (1984)presented their startling result that the fraction of blue galaxies inclusters increases with look-back time or redshift. This B–O effectseems to be confirmed by later studies (Couch & Sharples 1987;Rakos & Schombert 1995; Ellingson et al. 2001; Kodama & Bower2001b; Margoniner et al. 2001; Nakata et al. 2001; Fairley et al.2002; Goto et al. 2003b; Tran et al. 2005; but see also Fairley et al.2002; Andreon, Lobo & Iovino 2004). This effect suggests that starformation activities in cluster galaxies change with time, in the sensethat the fraction of star-forming galaxies decreases with time.

These three axes, that is, environment, mass, and time, must beclosely related to one another. For example, the morphology–densityrelation is found to evolve (Dressler et al. 1997; Fasano et al. 2000;Treu et al. 2003; Smith et al. 2005, but see also Andreon 1998).Tanaka et al. (2004) showed that the environmental dependence ofstar formation activity and morphology of galaxies changes as afunction of galaxy luminosity (mass). To untie this complexity, webase our analyses on panoramic imaging data of two high-redshiftclusters, CL0015.9+16 at z = 0.55 and RX J0152.7–13 at z = 0.83,

and the SDSS data. The former data are obtained as part of our on-going Subaru distant cluster project called PISCES. Our data spanwide ranges in environment, time, and stellar mass, and give us aunique opportunity to investigate star formation activity along allthe three axes simultaneously for the first time.

The structure of this paper is as follows. In Section 2, we brieflyreview our project PISCES and our data. Also we describe the dataof the local Universe that we use. Then we move on to the procedurewe adopt to eliminate foreground and background contamination inSection 3. Before presenting the main results, we summarize var-ious photometric and environmental quantities used in this paperin Section 4. We examine the environmental dependence of galaxystar formation in Section 5. Based on results obtained in Section 5,we redefine environments in Section 6. Colour–magnitude diagramsand luminosity functions in various environments are shown in Sec-tion 7 and 8, respectively. In Section 9, we discuss the implicationsof our results on the formation and evolution of galaxies. Finally, asummary is given in Section 10. Most of the important conclusionsin this paper are drawn from the results presented in Section 5.1,Section 7, and Section 9. Readers interested only in our primaryconclusions can go to these sections directly after a brief reading ofSections 2–4.

Throughout this paper, we assume a flat Universe with �M = 0.3,�� = 0.7 and H 0 = 70 km s−1 Mpc−1. We use the AB magnitudesystem for observed quantities and the Vega-referred system for rest-frame ones. We use the following abbreviations: CMD for colour–magnitude diagram, CMR for colour–magnitude relation, and LFfor luminosity function.

2 DATA

2.1 PISCES project

We are conducting a systematic study of cluster evolution basedon panoramic multiband imaging with the Subaru Prime FocusCamera (Suprime-Cam) and optical multislit spectroscopy with theFaint Object Camera And Spectrograph (FOCAS) on Subaru. In thissection, we briefly review the project Panoramic Imaging and Spec-troscopy of Cluster Evolution with Subaru (PISCES). The readershould refer to Kodama et al. (2005) for further details.

The primary aims of this project are two-fold: (1) to map outlarge-scale structures around distant clusters to trace the cluster as-sembly history, and (2) to look into galaxy properties in detail asa function of environment along the structures in order to identifydirectly the environmental effects acting on galaxies during their as-sembly to higher density regions. The unique feature of this projectis its wide-field coverage by taking advantage of the Suprime-Cam(Miyazaki et al. 2002), which provides a 34 × 27 arcmin2 fieldof view. Therefore, PISCES will provide an opportunity to linkan evolutionary path between galaxies in the local Universe andhigh-redshift counterparts over a wide range in environments. Also,the depth of PISCES, reaching down to M∗ + 4 at z ∼ 1 with an8-m telescope, will shed light on the nature of faint galaxies at highredshifts, which probably show quite different properties comparedwith massive galaxies, as seen in the local Universe (e.g. Baldryet al. 2004; Kauffmann et al. 2004). As part of this on-going project,we observed CL0015.9+1609 and RX J0152.7–1357 in 2003September.

The cluster CL0015.9+1609 (CL0016 for short) is one of the mostextensively studied galaxy clusters, and it has been observed in vari-ous wavelength ranges: X-ray (e.g. Worrall & Birkinshaw 2003), UV(Brown et al. 2000), optical (e.g. Dressler et al. 1999), and submm

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270 M. Tanaka et al.

(e.g. Zemcov et al. 2003). Koo (1981) suggested the existence of alarge-scale structure around the cluster based on photographic pho-tometry, and later, the cluster was found to have companion clusters(Hughes, Birkinshaw & Huchra 1995; Connolly et al. 1996; Hughes& Birkinshaw 1998). A clear CMR is seen (Ellis et al. 1997; Dahlenet al. 2004), and the cluster is known to have a very low blue fraction(Butcher & Oemler 1984).

The cluster RX J0152.7–1357 (RX J0153 for short) is one ofthe most X-ray luminous distant (z > 0.7) clusters known. Thecluster was discovered independently in the ROSAT Deep Clus-ter Survey (RDCS; Rosati et al. 1998) and in the Wide AngleROSAT Pointed Survey (WARPS; Scharf et al. 1997; Ebelinget al. 2000). Later, it was also detected in the Serendipitous High-redshift Archival ROSAT Cluster (SHARC) survey (Romer et al.2000). Since these discoveries, the cluster has been the subject ofBeppoSAX, Chandra and XMM–Newton observations (Della Cecaet al. 2000; Maughan et al. 2003; Jones et al. 2003). Observationsof the Sunyaev–Zeldovich effect (Joy et al. 2001), spectroscopicfollow-up (Demarco et al. 2005; Homeier et al. 2005; Jørgensenet al. 2005), near-IR imaging (Ellis & Jones 2004), and weak-lensing analysis (Huo et al. 2004; Jee et al. 2004) (Umetsu et al., inpreparation) have also been performed. All these studies show thatRX J0153 is indeed a very massive cluster at z = 0.83: a bolometricX-ray luminosity of >1 × 1045 ergs s−1, and a total dynamical massof ∼1 × 1015 M�. Based on the ACS weak-lensing analysis, Jeeet al. (2004) suggested that the previous cluster mass estimates maybe an overestimation.

Below, we briefly summarize our observation of these two clus-ters. Details of the observation and data reduction are described inKodama et al. (2005).

The observing conditions were excellent during the nights witha typical seeing size of 0.6 arcsec. CL0016 was observed in BVRi′ z′ and RX J0153 in V Ri′ z′. Exposure times and limiting mag-nitudes are shown in Table 1. Object detection is performed usingSEXTRACTOR (v.2.3.2; Bertin & Arnouts 1996). Objects in CL0016are i′-band selected and those in RX J0153 are z′-band selected. Itshould be noted that the i′-band for CL0016 and the z′-band forRX J0153 correspond to almost the same rest-frame wavelengthrange, and therefore we have little difference in selection effectsbetween the two clusters. We use MAG AUTO for total magnitudesand 2-arcsec diameter aperture magnitudes for colours. Magnitudezero-points determined by the photometric standard stars taken dur-ing the same observing run are found to be offset (�0.1 mag) withrespect to the stellar spectral energy distributions (SEDs) of Gunn& Stryker (1983). We shift the zero-points so as to match with Gunn& Stryker stars. Star–galaxy separation is performed on the basis ofFWHM versus total magnitude diagrams.

Table 1. Exposure times and limiting magnitudes (AB system). Limitingmagnitudes are shown as a 5σ limit measured in a 2-arcsec aperture.

Cluster Filter Exposure time (min) Limiting magnitude

CL0016 B 90 26.9V 96 26.2R 64 26.0i′ 60 25.9z′ 47.5 24.6

RX J0153 V 120 26.7R 116 26.5i′ 75 26.1z′ 77 25.0

In what follows, we restrict ourselves to i ′ < 24.5 galaxiesin CL0016 and z′ < 25.0 galaxies in RX J0153. These mag-nitude cuts ensure that we are not affected by incompletenesseffects.

2.2 SDSS

As a local counterpart of the two high-z samples, we use thedata from the Sloan Digital Sky Survey (SDSS; York et al. 2000;Stoughton et al. 2002; Abazajian et al. 2003, 2004). The SDSSobserves one-quarter of the sky both photometrically and spectro-scopically. The imaging survey is performed in five optical bands,u, g, r , i , and z (Fukugita et al. 1996; Gunn et al. 1998; Hogg et al.2001; Smith et al. 2002; Pier et al. 2003). The spectroscopic surveyis done with a pair of double fibre-fed spectrographs which covers3800–9200 Å. Each fibre subtends 3 arcsec on the sky. Since a fi-bre cannot be placed closer than 55 arcsec to a nearby fibre due tomechanical constraints, the tiling algorithm has been developed toreduce the number of unobserved objects (Blanton et al. 2003b).The overall completeness of the spectroscopic survey is expected tobe over 90 per cent.

We use the public data of the second data release (DR2;Abazajian et al. 2004). Galaxies in the Main Galaxy Sample (Strausset al. 2002) are used here. The sample covers 2627 deg2 of the sky.We extract galaxies at 0.005 < z < 0.065. We perform a volumecorrection to a flux-limited sample taking into account large-scalestructures, instead of making a volume-limited sample, so that wecan statistically reach as deep as the high-z samples in terms ofmagnitude relative to the characteristic magnitude. Details of thevolume correction are described in Appendix A. We use Petrosianmagnitudes (Petrosian 1976; Stoughton et al. 2002) for total mag-nitudes and model magnitudes for colours. All these quantities areGalactic extinction corrected (Schlegel, Finkbeiner & Davis 1998),and k-corrected using the code of Blanton et al. (2003c, v3 2).

Main galaxies in the SDSS are r-band selected (Strauss et al.2002), while those in RX J0153 and CL0016 are selected in therest-frame �g-band. To minimize selection effects, we apply themagnitude cut of g < 18 to our SDSS sample so that the sam-ple mimics a g-selected one at the cost of reducing the number ofsample galaxies. This cut leaves 41 695 galaxies. In summary,the galaxies in RX J0153, CL0016 and SDSS are all selected atsimilar wavelengths in the rest-frame nearly corresponding to theg-band.

3 C O N TA M I NAT I O N S U B T R AC T I O N

Since our two clusters from PISCES lie at high redshifts and ourimaging is deep, galaxies at the cluster redshifts are heavily con-taminated by foreground and background galaxies. Thus, in orderto study galaxies at the cluster redshifts, it is essential to elimi-nate the contamination. Our strategy for the contamination subtrac-tion is two-fold. First, we apply a photometric redshift techniqueto largely eliminate fore-/background galaxies (e.g. Kodama et al.2001a; Nakata et al. 2005). Although photometric redshift is a pow-erful tool, the contamination remains at a non-negligible level. Wetherefore statistically subtract the remaining contamination on thebasis of CMDs (Kodama et al. 2001a; Pimbblet et al. 2002). Eachprocedure is described in the following subsections.

3.1 Photometric redshifts

We apply the photometric redshift code of Kodama, Bell &Bower (1999) to the photometric catalogue of the PISCES clusters

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Figure 1. Left: photometric redshifts plotted against spectrally determined redshifts for RX J0153. In the inset, we show differences between photometricredshifts and spectroscopic redshifts as a function of R − z′ colour, which is close to the rest-frame U − V colour, using galaxies at 0.78 < z spec < 0.88. Right:same as the left-hand panel but for CL0016. In the inset, we use V − i ′ colour and galaxies at 0.5 < z spec < 0.6.

(Kodama et al. 2005). The photometric redshift utilizes the popula-tion synthesis model of Kodama & Arimoto (1997). Star formationhistories of model templates are described by a combination of theelliptical galaxy model with a very short time-scale (0.1 Gyr) of starformation and the disc model with a much longer time-scale of τ =5 Gyr (star formation rate ∝ e−t/τ ). The models are constructed so asto reproduce the observed colours of galaxies (Kodama et al. 1999).It is found that the observed colours of the CMR are slightly offsetcompared with the model colours. We shifted the model zero-pointssystematically so that the model colours of the CMR match with theobserved colours at the cluster redshifts. These shifts are requiredto calibrate and improve our photometric redshifts.

To assess the accuracy of our photometric redshifts, we comparethe photometric redshifts with spectroscopically determined red-shifts as shown in Fig. 1. For RX J0153, we use the spectroscopicsample of Jørgensen et al. (2004). Our photometric redshifts arefairly good for cluster galaxies. But, at z < 0.5, the accuracy israther poor since we lack U- and B-band data. Excluding the mostdeviant galaxy at z spec = 0.745, the mean and median of zphoto −z spec are +0.0067 and +0.0093, respectively. The standard devia-tions around the mean and median are found to be 0.045 and 0.047.For CL0016, we compile spectroscopic redshift data of Hugheset al. (1995), Munn et al. (1997), Hughes & Birkinshaw (1998), andDressler et al. (1999). Excluding the two most deviant galaxies atz spec ∼ 0.4, the mean and median of zphoto − z spec are +0.0036 and−0.0118, respectively. The standard deviations around the meanand median are 0.055 and 0.057. The figures demonstrate that ourphotometric redshifts are fairly accurate. We note, however, that thespectroscopic samples are heterogeneous, and the face values quotedabove should not be overinterpreted. Note as well that the photomet-ric errors of red galaxies at our magnitude limit are typically σ (V )= 0.1, σ (R) = 0.06, σ (i ′) = 0.04, σ (z′) = 0.06 in RX J0153 andσ (B) = 0.09, σ (V ) = 0.05, σ (R) = 0.02, σ (i ′) = 0.02, σ (z′) = 0.04in CL0016 (errors of blue galaxies are smaller than these values).Thus, photo-z should work with a reasonably good accuracy (|�z|< 0.1) even at our magnitude limits (Kodama et al. 1999).

Using cluster galaxies only (0.78 < z spec < 0.88 for RX J0153 and0.50 < z spec < 0.60 for CL0016), we examine the accuracy of pho-tometric redshifts as a function of galaxy colour. For RX J0153, wecannot identify any bias in the photometric redshifts, although welack galaxies having intermediate colours (R − z′ ∼ 1.4). We find,however, that we tend to underestimate redshifts for galaxies withintermediate colours (V − i ′ ∼1.5) by�z ∼−0.1 in CL0016. A pos-sible reason for this colour dependence is that slightly bluer galaxiesat cluster redshift than those on the CMR tend to be regarded as redgalaxies at slightly lower redshifts because of the colour–redshiftdegeneracy (Kodama et al. 1999). But, we need more spectroscopicdata to address this issue fully. Our photometry covers a similarrest-frame wavelength regime for both RX J0153 and CL0016, andwe expect that we also tend to underestimate redshifts of galaxieswith intermediate colours in RX J0153. In both samples, photomet-ric redshifts of red galaxies are found to be fairly accurate, thougha small offset zphoto−z spec = +0.02 is seen.

Galaxies outside of a certain photometric redshift range aroundthe cluster redshift are regarded as fore-/background galaxies. Thechoice of the redshift range is a trade-off between the complete-ness of cluster galaxies and the contamination of fore-/backgroundgalaxies. If we adopt a small redshift range, the contamination willbe reduced, but a price to be paid is a selection bias towards redgalaxies. If a wide redshift range is adopted, the selection bias willbe reduced at the cost of increasing an amount of the contamination.Since we are interested in galaxy properties, especially colours, weprefer to construct an unbiased sample. For this purpose, we adoptz cl − 0.12 < zphot < z cl + 0.05, where zcl is a cluster redshift.This selection criterion will eliminate the colour selection bias well,while maintaining the amount of the contamination to be minimal(Fig. 1). To be specific, we adopt 0.42 < zphot < 0.60 for CL0016and 0.71 < zphot < 0.88 for RX J0153.

It should be noted that these redshift ranges are different fromthose adopted in Kodama et al. (2005). They adopted narrower red-shift ranges to enhance large-scale structures by tracing, primarily,red galaxies.

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3.2 Statistical contamination subtraction

Even though the photometric redshift is effective at reducing thefore-/background contamination, contamination still remains at anon-negligible level within our photometric redshift ranges. Weconstruct a control field sample and statistically subtract such re-maining contamination on the basis of the galaxy distribution onCMDs. Note that we do not apply any contamination subtraction inthe SDSS sample since galaxies are spectroscopically observed.

Here we describe only the essence of the procedure of the statis-tical subtraction. See Appendix B for details. We adopt a modifiedprocedure of Kodama & Bower (2001b) and Pimbblet et al. (2002).In brief, the distribution of control field galaxies on the CMD isused as a probability map of the contamination. The control fieldsample is defined as low-density regions in each field of RX J0153and CL0016, as shown later. The choice of the control field is ar-bitrary, but our results do not strongly rely on a particular choiceof the control field. We select two separated regions in each fieldto minimize cosmic variance. The average galaxy density in the se-lected regions is denoted as � control. The subtraction of the remainingcontamination is done on the CMD using a Monte Carlo method.We statistically subtract the galaxy distribution on the CMD of thecontrol field from that of the target field.

3.3 Concerns about the contamination subtraction

Although we carefully subtract the contamination, uncertaintiesarising from the subtraction cannot be ignored. In this subsection, webriefly address the robustness of our conclusions presented below.

In the following sections, we discuss the fraction of red galaxies.As shown above, photometric redshifts of red galaxies are moreaccurate than those of blue galaxies, and we may tend to missblue galaxies. Therefore, the real fraction of red galaxies wouldbe smaller than we observe. This will strengthen our conclusions,the observed density dependence of galaxy colours and evolutionarytrends. We cannot, however, evaluate the amount of missing bluegalaxies with the data at hand.

Errors in the statistical subtraction are also a concern (e.g.over/under subtraction). However, taking advantage of the wide fieldof view, we can take wide areas for our control field sample, namely∼270 arcmin2 and ∼210 arcmin2 in RX J0153 and CL0016, respec-tively. Therefore cosmic variance must be averaged to some extentand is not expected to be a serious problem. We repeated MonteCarlo runs of statistical subtraction many times and confirmed thattrends we discuss later are seen in most realizations. In other words,we discuss secure results only. Although the robustness of our con-tamination subtraction must be confirmed later spectroscopically, itis unlikely that our results significantly suffer from uncertainties inthe contamination subtraction.

4 D E F I N I T I O N S O F E N V I RO N M E N TA LPA R A M E T E R S A N D D E R I VAT I O NO F R E S T- F R A M E QUA N T I T I E SA N D S T E L L A R M A S S

Before presenting our results, we summarize various photometricand environmental parameters used in this paper. We begin by in-troducing environmental parameters used to characterize local andglobal environments. We then describe galaxy properties such asmagnitude, colour, and stellar mass.

4.1 Environmental parameters

4.1.1 Local density

In this paper, we use a nearest-neighbour density to characterize theenvironment mainly because it is a frequently used indicator andcomparisons with other studies can be made directly. In RX J0153and CL0016, all the galaxies in the selected photometric redshiftrange are projected on to the redshift of the main cluster, and densityis estimated from the distance to the tenth-nearest galaxy from thegalaxy of interest and is denoted as �10th. We use a circular aperturein the density calculation. This density is called the local densityhereafter. It should be noted that local density is actually a surfacedensity. Galaxies that reach the edge of our field of view beforefinding their tenth-nearest galaxies are not used in the analysis sincethe local density of such galaxies is not correctly estimated. The localdensity for the two high-z clusters is calculated in both physical andcomoving scales.

In the SDSS, local density is estimated in a similar manner asBalogh et al. (2004b). In brief, galaxies within ±1000 km s−1 inthe line-of-sight velocity space from the galaxy of interest are pro-jected on to the redshift of the central galaxy, and local density isdefined from the distance to fifth-nearest neighbour. When count-ing galaxies, we use only those brighter than MV = −19.5, whichis volume-limited out to z = 0.065 (our maximum redshift range).This density is denoted as �5th. Although we use the distance tothe fifth-nearest neighbour, our results are essentially unchangedif we use the tenth-nearest neighbour as used for RX J0153 andCL0016. Since the contamination of fore-/background galaxies isvery small in the SDSS sample, we prefer to adopt a smaller numberso that the density represents more ‘local’ environments. The fibre-collision problem may affect our density estimates in high-densityenvironments. However, the effect is not significant (Tanaka et al.2004). Local density for the SDSS is calculated in the physical scaleonly. There is, however, little difference between the physical andcomoving density since galaxies lie at very low redshifts.

4.1.2 Global density

Global density is defined as the surface galaxy density in a fixedradius of 2 Mpc around the central galaxy. Galaxies are projected onto the cluster redshift (RX J0153 and CL0016) or on to the redshift ofthe galaxy (SDSS) in question, in just the same manner as the localdensity. Since we aim to characterize the global environment, globaldensity is evaluated as a comoving density. By combining local den-sity with global density, we can separate poor groups from richclusters quantitatively. The effectiveness of this method is describedin Appendix D. In what follows, global density is denoted as �global.

4.2 Magnitudes, colours and stellar masses

4.2.1 Rest-frame magnitudes and colours

For RX J0153 and CL0016, we derive rest-frame absolute V-bandmagnitude and U − V colour from the observed magnitudes andcolours using the model templates of the photometric redshift code(Kodama et al. 1999). The conversion applied in the observed quan-tities to derive the rest-frame quantities is small. Within the selectedredshift ranges, the effect of galaxy evolution is estimated to be�MV < 0.2 and �(U − V ) < 0.05 for both RX J0153 and CL0016.As for the SDSS sample, we estimate the rest-frame V and U − Vusing the code by Blanton et al. (2003c, v3 2). We recall that, for

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Table 2. The V limiting magnitudes (Vega)and the limiting stellar masses of the threesamples.

Sample M V,lim M ∗,lim/M�SDSS −17.5 4 × 109

CL0016 −18.0 5 × 109

RX J0153 −18.0 4 × 109

conventional reasons, we use the Vega-referred system in the rest-frame V magnitude and the U − V colour. Table 2 summarizes thelimiting magnitudes for the three samples.

4.2.2 Stellar masses

We derive approximate stellar masses of galaxies. For each sample,the stellar mass-to-light ratio (M ∗/L) and M∗ are derived fromthe model templates used in the photometric redshift code. OurL is defined at the rest-frame ∼V-band, and so the estimates ofM∗ are affected by on-going/near-past star formation activities. It isexpected that our stellar mass estimates are relatively accurate for redgalaxies, since red galaxies are not actively forming stars. However,stellar masses of blue galaxies cannot be reliably determined. Wefind that M ∗/L ratios span a factor of ∼4 depending on the colourof the galaxies, and we can get only rough estimates in stellar massalthough we correct for the mass-to-light ratio based on the colours[spectral energy distribution (SED) fitting]. Moreover, stellar massis a model-dependent quantity, since the mass-to-light ratio dependson the stellar initial mass function (IMF). Our M∗ estimate is basedon the Kodama & Arimoto (1997) population synthesis model, andwe assume the IMF of x = 1.10 for the elliptical models and x =1.35 for the disc models with the mass range of 0.1–60 M�. Thelimiting stellar masses that we can trace completely are shown inTable 2.

5 C O L O U R – D E N S I T Y R E L AT I O N S

We apply the photometric redshift technique and discover large-scale structures around both RX J0153 and CL0016 clusters. Detailsare described in Kodama et al. (2005). Due to the wide-field cover-age of the Suprime-Cam, we obtain a wide variety of environments,i.e. sparse fields, poor groups, and rich clusters. Here we exam-ine the relationship between galaxy colours (star formation rates)and environments. We stick to local environments in this section.The effects of global environments are examined later. First, wecharacterize the environment by surface galaxy density (local den-sity). Next, the environment is defined on the basis of surface massdensity.

5.1 Dependence on surface galaxy density

Based on data from large surveys of the local Universe, Lewis et al.(2002) and Gomez et al. (2003) showed that galaxy star formationbegins to decline sharply, in a statistical sense, at a certain local den-sity. In what follows, this sharp decline is referred to as a ‘break’, andthe density where the ‘break’ is seen is referred to as the break den-sity. Non-star-forming galaxies dominate regions above the breakdensity, whereas star-forming galaxies are the dominant populationbelow the break density. Based on the data from the SDSS, Tanakaet al. (2004) showed that the break is seen only for galaxies fainter

than M∗r + 1, and brighter galaxies show no clear break in the local

Universe. Following this work, we examine the environmental de-pendence of star formation for bright and faint galaxies separately.Galaxies brighter than M∗

V + 1 are defined as bright galaxies, andthose fainter than that limit are defined as faint galaxies (the valueof M∗

V at each redshift is given in Section 8).Fig. 2 shows our results. In the figure, the U − V colour is cor-

rected for the slope of the CMR so that galaxies on the CMR havethe same colour as that of a M∗

V galaxy and is denoted as (U −V )corr (i.e. CMR is transformed into a horizontal sequence). Thefaint galaxies (solid line) show a break, a prominent change in their(U − V )corr colours, at the densities marked by the dot-dashed linesin the figures. It is interesting that the break in galaxy colours isseen at all redshifts examined here. In contrast, the bright galaxies(dashed line) do not show such a strong break, especially the medianlines. The (U − V )corr colours of bright galaxies are systematicallyredder than those of faint galaxies at any local density.

In order to quantify the break feature, we measure the slope of thecolour–density relation using the median lines in Fig. 2 as a functionof local density for bright and faint galaxies separately. The resultis shown in Fig. 3. The faint galaxies show a strong change in slopeat the densities shown by the dot-dashed lines, while bright galaxiesshow a much weaker change there. Here we define the break densityat which we see the strong colour change in faint galaxies, as markedin Figs 2 and 3.

We change the threshold magnitude used for separatingbright/faint galaxies and check how the presence of the break den-sity of each cluster depends on the magnitude of galaxies. It is foundthat galaxies brighter than M∗

V + 1 do not show a prominent break,while those fainter than that limit show a break at the same density.

There are two possible effects that cause the break. One is thatthe fraction of red galaxies relative to blue galaxies begins to increaseabove the break density. The other is that blue galaxies becomesystematically redder (but bluer than galaxies on the CMR) above thebreak density. These possibilities are investigated in the right-handpanels of each plot in Fig. 2. Red and blue galaxies are separated at(U − V )CMR − 0.15. In RX J0153 and CL0016, the (U − V )corr ofblue galaxies does not change with density, while the fraction of redgalaxies strongly changes with density. This means that the break iscaused by the change in the population fraction of red galaxies. Asfor the SDSS plot, the (U − V )corr of faint blue galaxies becomessystematically redder above the break density. On the other hand,the (U − V )corr of bright blue galaxies does not strongly change withdensity. The fraction of red galaxies is found to depend strongly ondensity. Therefore, we conclude that, in the SDSS plot, the break iscaused by both effects.

The break densities are used to define environments in Section 6.We note that a small change in the break density has no significanteffect on our results. Note as well that the break density at z = 0.55and z = 083 is 3–5 times higher than the control field density, andthus the uncertainty in the statistical contamination subtraction isnot a concern. As described in Section 3.3, photometric redshiftsmay miss a fraction of blue galaxies. This will strengthen the ob-served break since the break is primarily driven by the change inthe population fraction of red galaxies relative to blue galaxies.Further discussion on the break density of each cluster is made inSection 9.2.

5.2 Dependence on surface mass density

Gray et al. (2004) first presented the environmental variation ofgalaxy colours as a function of surface mass density. Inspired by

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Figure 2. The colour–density relation in RX J0153 (top left), CL0016 (top right) and SDSS (bottom). These plots show one realization of our Monte Carlorun for the statistical contamination subtraction. Note that no contamination subtraction is performed in the SDSS plot. Left-hand panel in each plot: the (U− V )corr colour plotted against local density. The lines show the median and 25th percentile of the distribution of bright (dashed line) and faint (solid line)galaxies as noted in the RX J0153 panel. The median line is associated with bootstrap 90 per cent intervals as shown by the error bars. Densities are expressedin the comoving density (top ticks) and in the physical density (bottom ticks) and are contamination corrected (i.e. shifted leftward by �control). The verticallines show the break density of each cluster. The arrow indicates � control, where a half of galaxies are statistically subtracted. In the SDSS plot, local densityis shown in a physical scale only. For clarity, one-tenth of all the SDSS galaxies are randomly selected and plotted. Each bin contains 100 bright/200 faintgalaxies in RX J0153 and CL0016, and 1000 bright/2000 faint galaxies in the SDSS plot. Top right-hand panel in each plot: the (U − V )corr colour plottedagainst local density for blue [U − V < (U − V )CMR − 0.15] galaxies. The lines show the median of the distribution of bright/faint galaxies as noted in theRX J0153 panels. The associated error bars are the bootstrap 90 per cent intervals. The horizontal line means (U − V )CMR − 0.15. The vertical line showsthe break density of each cluster. Bottom right-hand panel in each plot: the fraction of red galaxies plotted against local density. The meanings of the lines aregiven in the RX J0153 panels. The errors are based on Poisson statistics.

this work, we discuss colours of galaxies as a function of surfacemass density in this subsection. This investigation is particularlyinteresting since surface galaxy density, on which our analysis inthe previous subsection is based, represents the density of luminousmatter around the cluster, while the surface mass density estimatedvia the weak-lensing analysis represents the total mass includingdark matter. These two densities do not necessarily agree with eachother, and comparisons between the dependence on galaxy densityand that on mass density will give us a hint of physical mechanismsthat affect galaxy properties.

The weak lensing mass reconstruction of RX J0153 is describedin detail in Umetsu et al. (in preparation). The lensing convergenceκ is related to the surface mass density �κ by κ(θ) = �κ (θ)/�eff

κ,crit.As a fiducial value, Umetsu et al. (in preparation) adopted �eff

κ,crit =3.1 × 1015 h70 M� Mpc−2.

We plot in the left-hand panel of Fig. 4 the surface galaxy densityagainst the lensing convergence κ(θ). A positive correlation is found

between κ and galaxy surface density, especially at high densitieswith κ � 0.1. On the other hand, no clear correlation can be seen inthe low-density regime.

In the right-hand panel, the (U − V )corr colour is plotted againstκ . Although it is not as clear as in Fig. 2, there is a hint of a breakin the (U − V )corr colour at κ ∼ 0.1. This threshold corresponds to�κ ∼ 3 × 1014 M� h70 Mpc−2 in physical units. Gray et al. (2004)found a similar break at the surface mass density of �κ ∼ 3.6 ×1014 M� h70 Mpc−2, which is consistent with our estimate. How-ever, this apparent threshold density of κ ∼ 0.1 is comparable to therms noise level in the reconstructed κ map, σ κ � 0.10. Therefore,the underlying mass density threshold can be smaller than what weobtained, κ ∼ 0.1. It is therefore premature to say if galaxy proper-ties are more strongly related to galaxy density than to mass density.We note that Jee et al. (2004) recently reported that the distributionof galaxies, mass, and intracluster medium are all different in thiscluster, suggesting on-going cluster merger.

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Figure 3. The slope of the colour–density relation [�(U − V )/�log �]as a function of local density along the median loci in Fig. 2. Thetop/bottom ticks are comoving/physical densities as in Fig. 2. The panelsshow RX J0153, CL0016, and SDSS (from left to right). The filled/opensymbols show bright/faint galaxies, respectively. The vertical dot–dashedline means the break density. The errors are estimated from the bootstrapresampling.

Figure 4. Left: relationship between the surface galaxy density (local den-sity) and normalized surface mass density (κ) in RX J0153. The lines showthe median and quartiles (25 per cent and 75 per cent) of the distribution. Thehorizontal dashed line means the break surface galaxy density. The arrowindicates κ = 0.1 where S/N of the mass map is unity. Right: the (U − V )corr

colour plotted against κ . The meanings of the lines are the same as Fig. 2.The arrow indicates mass density of S/N = 1.

Although we cannot draw a firm conclusion on this analysis, thisis potentially an interesting way of investigating the environmentaldependence of galaxy properties. If, for example, galaxy–galaxyinteractions are the main driver of the environmental dependence,we expect to see a stronger dependence on galaxy density than onmass density. On the other hand, if interactions with the clusterpotential are the main driver, we expect a stronger dependence onmass density.

6 D E F I N I T I O N S O F F I E L D , G RO U PA N D C L U S T E R E N V I RO N M E N T S

In order to address galaxy properties as functions of environmentand time, the definition of the environment must be quantitative andapplicable at all redshifts. For this reason, we define environmentsby galaxy density. For analyses in the following sections, we definethree environments: field, groups and clusters as shown in Figs 5, 6and 7.

We do not examine galaxies in lower density regions than � control

because most of them are considered to be fore-/background galax-ies. We found in the previous section that red galaxies dominateenvironments denser than the break density. Motivated by this, weuse the break density of each cluster to define the environments.Galaxies in higher density regions than � control, but in lower densityregions than the break density (�break) are considered to be in thefield environment (i.e. � control < � local < �break).

Galaxies in higher density regions than �break belong to groupsand clusters. We now aim to investigate the effects of the globalenvironment: are there any differences between group galaxies andcluster galaxies? This question is particularly interesting since dif-ferent physical mechanisms are effective in different environments.For example, ram-pressure stripping of cold gas (Gunn & Gott 1972;Abadi, Moore & Bower 1999; Quilis, Moore & Bower 2000) and ha-rassment (Moore et al. 1996, 1999) are expected to play a role only inrich clusters. Low-velocity galaxy–galaxy interactions (e.g. Mihos& Hernquist 1996) are expected to be effective in galaxy groups.Strangulation (which is often referred to as suffocation, starvationor halo gas stripping) is considered to be effective both in groupsand clusters (Larson, Tinsley & Caldwell 1980; Balogh, Navarro& Morris 2000; Diaferio et al. 2001; Okamoto & Nagashima2003).

The global density defined in Section 4.1.2 is found to work wellin separating groups from clusters. The global density traces thegalaxy density over a large scale, and thus the global density is agood measure of the richness of galaxy systems. We define a globallydenser environment than the break density as clusters and globallyless dense environment as groups. That is, clusters are defined by� local > �break and �global > �break, and groups by � local > �break

and �global < �break. As illustrated in Figs 5, 6 and 7, our separationof environments is reasonable. We show in each plot the virial radiusof the cluster. The virial radius of RX J0153 is taken from Maughanet al. (2003), and those of CL0016 and SDSS are evaluated fromvelocity dispersions using the recipe of Carlberg, Yee & Ellingson(1997). As seen in the plots, the break density corresponds to theoutskirts of clusters. This is quantitatively consistent with that seenin the local Universe where the break density corresponds to oneto two Rvir (Gomez et al. 2003; Tanaka et al. 2004). We note thatthe break density also corresponds to a density typical of isolatedgroups.

We are aware that, since we define the environment based ongalaxy colours, we may obtain ‘biased’ environmental dependen-cies of galaxy colours. For example, since we define cluster envi-ronment where red galaxies are abundant, clusters are, by definition,dominated by red galaxies. However, the fact that the break densitycorresponds to the outskirts of clusters is a good justification of ourdefinition. Therefore, we consider that results presented below arenot biased products of the environment definition.

To sum up this section, we tabulate the definitions of environmentsin Table 3. Based on these environments, we look into CMDs andLFs of galaxies in the following sections.

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Figure 5. Galaxy distribution in RX J0153. North is up. The filled rectangles and open circles show cluster and group galaxies. The large dots are field galaxies.Galaxies in the lower density regions than � control are shown by the small dots. Galaxies in the large dashed rectangles are used as control field galaxies, andthey are used in the statistical contamination subtraction procedure. Galaxies too close to the edge of our field of view are not plotted because they have no localdensity estimates. The top and right ticks show the comoving scales in unit of Mpc. The circle shows Rvir (Maughan et al. 2003). No statistical contaminationsubtraction is applied in this plot.

7 C O L O U R – M AG N I T U D E D I AG R A M S

Galaxies in clusters are known to show a tight CMR (e.g. Bower,Lucey & Ellis 1992). This relation is observed up to z = 1 and evenbeyond (Kodama et al. 1998; Stanford, Eisenhardt & Dickinson1998; Nakata et al. 2001; Blakeslee et al. 2004; De Lucia et al.2004; Lidman et al. 2004). In this section, we aim to investigatethe CMDs with particular attention to the group and field environ-ments, which have not been intensively studied yet especially at highredshifts.

Fig. 8 shows the rest-frame CMDs. We estimate the slope andscatter of the CMRs based on an iterative 2σ -clipping least-squaresfit and the results are shown in Figs 9 and 10. Note that the CMRslopes are measured from galaxies brighter than M∗

V + 2 which showa tight relationship, while the scatter around the CMR is measuredfor galaxies down to our magnitude limits. Errors are estimated fromthe bootstrap resampling of the input catalogues, and the statisticalfield subtraction is performed in each run. Since our photometricredshift cuts have some ranges (�z ∼ 0.18), a projection effect is aconcern which may apparently enhance the intrinsic colour scatter.However, the colour difference in passively evolving galaxies withinthese redshift ranges are only 0.05 mag at most for both CL0016and RX J0153 (Kodama et al. 1998), therefore this effect is smallcompared to the amount of scatter we discuss here (>0.1 mag). Wesee five very interesting trends in these figures.

First, we find probable onset of the build-up of the CMR in fieldregions. In the field environment in RX J0153 at z = 0.83, we cannot

identify any clear CMR, although a clump of bright red galaxies canbe seen. In fact, the scatter around the CMR is very large (Fig. 9),being consistent with no CMR. Interestingly, however, a clear re-lation is seen in the same environment in CL0016 at z = 0.55 andalso in SDSS at z = 0 particularly at the bright end. We find that thescatter around the CMR at the bright end decreases from z = 0.83down to z = 0, while the scatter at the faint end does not decreaseclearly. That is, the bright end of the CMR is built-up with time,while there is no clear CMR at the faint end even at z = 0.

Secondly, we find an on-going build-up of the CMR in groups.Groups in RX J0153 show a CMR, but we find that the CMR isgetting weak at MV > −20. There are a number of red galaxiesat MV > −20, but they do not form a tight relation. The scatteraround the CMR shown in Fig. 9 reflects this visual impression. Atlower redshifts, group galaxies show a clear relation down to themagnitude limit. This leads us to suggest that we are witnessingthe build-up of the CMR in groups. An emerging picture is that theCMR grows from the bright end to the faint end, and not the oppositeway.

Thirdly, no such on-going build-up of the CMR is seen in clus-ters. We cannot see dramatic growth of the CMR in clusters. Clustergalaxies show a clear CMR down to the magnitude limit at all red-shifts considered here and the scatter around the CMR is alreadysmall at high redshifts.

One may suspect that the observed build-up of the CMR is anartefact: e.g. photometric redshifts miss a fraction of red galaxies inRX J0153, and it mimics the CMR build-up. But, this is not the case.

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Figure 6. Same as Fig. 5, but for CL0016. North is to the left.

Figure 7. Same as Fig. 5, but for SDSS. Only a patch of the sky is shown. Galaxies are thinly populated compared with Figs 5 and 6, but the magnitude limithere is shallower by ∼1 mag. The dots are field galaxies.

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Table 3. Definitions of environments. � control, � local,�break and �global are control field density, local density(�5th or �10th), break density and global density.

Environment Definition

Field � control < � local < �break

Group � local > �break and �global < �break

Cluster � local > �break and �global > �break

The accuracy of photometric redshifts does not depend on environ-ment. Accordingly, if there were a CMR in the field environmentsof RX J0153, it would have been found because we see the clearCMR in the cluster environments. The same argument can be ap-plied to the faint end of the CMR in groups. It should be noted thatour group environment is a composite of several individual groups,and hence a variation in group properties is not a major concern.Note as well that photometric redshifts are reliable for red galaxies(see Fig. 1). Therefore, we suggest that the observed build-up isreal.

Fourthly, the slope of the CMR does not change with environ-ment. The slope of the CMR in the field environment is found to besimilar to those in groups and clusters. This is quantified in Fig. 10.There is no convincing evidence that the CMR slope depends on en-vironment. Note that galaxy colours are measured within differentaperture sizes at different redshifts: Petrosian aperture in SDSS and2-arcsec aperture in CL0016 and RX J0153. CMR slopes changewith aperture sizes (Bower et al. 1992), and we do not discuss theevolution of the CMR slope here. Although the CMR seems to be

Figure 8. The rest-frame CMDs (U − V versus MV ) in RX J0153 (left), CL0016 (middle) and SDSS (right) are plotted. In each plot, the panels show differentenvironments, namely, field, group and cluster. Small dots mean statistically subtracted galaxies (except the SDSS plot). The solid line represents the CMRshifted blueward by 0.15 mag. The CMR is based on the model prediction of Kodama & Arimoto (1997). This solid line separates red and blue galaxies usedin the following analysis. The vertical dashed line is the magnitude limit, and the slanted dashed line shows the 5σ limiting colour. The arrows show M∗

V + 1,which is used to separate bright galaxies from faint galaxies in Section 5. In the cluster plot, the curved dashed lines show loci of stellar masses of 1010

and 1011 M�. Note that, in the SDSS plot, the volume correction is applied and the corrected galaxies are artificially plotted. Note as well that no statisticalcontamination subtraction is performed in the SDSS plot.

built-up at different epochs in different environments, this similarityof the CMR slope may be expected if the slope is primarily caused bythe metallicity effect rather than the age effect (Kodama & Arimoto1997; Stanford et al. 1998). The observed CMR build-up suggeststhat the slope of the CMR does not change after the formation epochof the CMR.

Fifthly, we confirm bimodality in galaxy colours. Galaxy prop-erties are known to have a strong bimodal distribution in the localUniverse (Strateva et al. 2001; Blanton et al. 2003a; Kauffmannet al. 2003; Baldry et al. 2004; Balogh et al. 2004b; Kauffmannet al. 2004; Tanaka et al. 2004). This bimodality is seen up to z ∼ 1(Bell et al. 2004). We confirm the bimodality although it becomesless clearly seen at higher redshifts in all the environments. Thebimodality is particularly noticeable in the field regions of CL0016and SDSS. High-density environments lack blue galaxies.

The highlight of this CMR analysis is that we observe the build-up of the CMR. It seems that the evolutionary stage of the CMRbuild-up is different in different environments: the cluster CMRis built first, and the CMRs of the group and field regions arebuilt later on. Another interesting implication is that the brightend of the CMR appears first, and the faint end is filled up lateron. We will further quantify and discuss the observed build-up inSection 9.3.

8 L U M I N O S I T Y F U N C T I O N S

The luminosity function (LF) is one of the most fundamental mea-sures of galaxy properties. This section studies LFs as functions ofenvironment and time. An LF is fitted by the Schechter function

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Figure 9. The colour dispersions around the CMR in �(U − V ). Themeanings of the symbols are shown in the middle panel. The error bars areestimated from the bootstrap resampling of input catalogues.

Figure 10. The slopes of CMRs plotted against the look-back time. Thehorizontal dashed line is a prediction of the Kodama & Arimoto (1997)model. The meanings of the symbols are shown in the panel. The error barsare estimated from the bootstrap resampling.

(Schechter 1976) of the form

φ(L) dL = φ∗(

L

L∗

)α

exp

(− L

L∗

)d

(L

L∗

), (1)

or equivalently, per unit magnitude,

φ(M) dM = 0.4 ln 10 × φ∗10−0.4(M−M∗)(α+1)

× exp[−10−0.4(M−M∗)

]dM . (2)

There are three parameters in the Schechter function: the normal-ization factor φ∗, the characteristic luminosity/magnitude L∗/M∗,and the faint end slope α. Best-fitting parameters are searched viathe conventional χ 2-minimizing statistics. Galaxies in RX J0153and CL0016 are binned into 0.5-mag steps, and those in the SDSSare binned into 0.25–0.5 mag steps. First, we examine total (= red+ blue) LFs at each redshift. Then we look into red/blue LFs indifferent environments and at different redshifts.

8.1 Total luminosity functions

Based on the photo-z selected samples of RX J0153 and CL0016and the spec-z sample of SDSS, the total LF of galaxies at eachredshift is constructed and shown in Fig. 11. Note that no statisticalcontamination subtraction is performed here.

The Schechter function gives a good fit (χ2ν ∼ 1) to the total LF

for RX J0153 and CL0016, but it is not a good fit for SDSS (χ 2ν ∼

11). The observed SDSS LF deviates from the Schechter functionat the faint end, and this deviation decreases the overall goodnessof fit. The deviation is possibly due to increasing contribution ofdwarf galaxies. We fit the Schechter function using galaxies withMV < −18.5 and obtain M∗

V = −21.12 and α = −1.01 with χ2ν ∼

4. We recall that the M∗V derived in Fig. 11 were used in separating

bright/faint galaxies in Section 5. Note that a small error in M∗V has

little effect on the results obtained in that section.Fig. 11 clearly shows that galaxies fade with time: M∗

V = −22.05at z = 0.83, M∗

V = −21.61 at z = 0.55, and M∗V = −21.24 at z = 0.

This observed fading, �M∗V = 0.81 mag from z = 0.83 to z = 0 and

�M∗V = 0.37 mag from z = 0.55 to z = 0, is consistent with a passive

evolution model (Kodama & Arimoto 1997; zf = 5), �MV = 0.80mag and 0.55 mag, respectively, within the errors. This suggests thatM∗

V is primarily determined by passively evolving galaxies. In ourcosmology, z = 0.83 and z = 0.55 correspond to a look-back timeof 7.0 Gyr and 5.4 Gyr, respectively.

The faint end slope α also seems to evolve. The number of faintgalaxies relative to bright galaxies increases with time, α = −0.94at z = 0.83 and α = −1.12 at z = 0. But the local value should betaken with caution since α is found to depend on the magnitude range

Figure 11. Total LFs in RX J0153 (top), CL0016 (middle) and SDSS (bot-tom). Note that no contamination subtraction is performed here. The left-hand panels show the error ellipse for each LF. The solid and dashed contoursshow χ2

ν = χ2ν,best + 1 and χ2

ν = χ2ν,best + 2, where χ2

ν,best is the χ2ν of the

best fit. The right-hand panels show LFs along with the best-fitting Schechterfunctions. The error bars are based on the Poisson statistics only.

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Figure 12. Rest-frame LFs in various environments in RX J0153 (top left), CL0016 (top right) and the SDSS (bottom). The left-hand panels in each plot showthe error ellipses (χ2

ν = χ2ν,best + 1) of the Schechter fits. The solid and dashed lines represent red and blue galaxies defined in Fig. 8. The filled points and

open squares show the best-fitting parameters for red and blue LFs, respectively. The right-hand panels show LFs. The filled and open symbols represent redand blue galaxies. The solid and dashed lines, respectively, show the best-fitting Schechter functions for red and blue galaxies. The statistical contaminationsubtraction is carried out here.

involved in the Schechter fit. The contribution of dwarf galaxies islikely to be significant at the faint end. We therefore do not try todraw any firm conclusion on the evolution of α.

In the RX J0153 and CL0016 fields, we have rich clusters and anumber of groups, and the contribution of clusters and groups to thetotal LF is larger compared with the SDSS LF. In the SDSS sample,the field, group and cluster galaxies comprise 80, 13 and 7 per centof the total number of galaxies. We scale the relative fraction offield, group, and cluster galaxies in RX J0153 and CL0016 to theSDSS values, and find that the derived Schechter parameters showonly a small change: �M∗

V = 0.1 and �α = 0.02. Thus, a differentenvironmental mix of galaxies does not strongly change our results.

8.2 Luminosity functions of red/blue galaxies

The fact that galaxy properties have strong bimodality in their dis-tribution motivates us to investigate LFs of red and blue galax-ies separately. We define red galaxies as those having U − V >

(U − V )CMR − 0.15. Blue galaxies are defined as those bluer thanthis limit. This definition is illustrated in Fig. 8. It should be notedthat, in the following, the statistical contamination subtraction isperformed on the LF bin – LF bin basis. This is different from theMonte Carlo approach that we adopt in the previous sections. Our

results are presented in Fig. 12. A general trend is that red galaxieshave a decreasing or flat faint end slope (though exceptions can befound), while blue galaxies have an increasing slope at any redshift.Some SDSS data points at the faint end deviate from the best-fittingSchechter function. This is because the SDSS data at the faint endhave small weights in the Schechter fitting due to the statisticalnature of the volume correction.

Now, we focus on the LFs of red galaxies. At z = 0, the faintend slope becomes gradually less steep in a denser environment.Hogg et al. (2003) reported, based on SDSS data, that faint redgalaxies preferentially populate in high-density environments. Asimilar conclusion was reached by De Propris et al. (2003) basedon the 2dF data. It is likely that red faint galaxies are selectivelylocated in high-density environments. A similar trend can be seenin RX J0153, but it is not statistically significant given the largeerror ellipses. Kajisawa et al. (2000), Nakata et al. (2001), Kodamaet al. (2004), Toft et al. (2004), and De Lucia et al. (2004) reportedthe deficit of red faint galaxies in high-redshift (z > 0.7) clusterscompared with local clusters. Although we cannot confirm the trendin the Schechter parameter, we do see the deficit of red faint galaxiesin the giant-to-dwarf ratio, which we will discuss in the next section.As for blue LFs, there is no convincing evidence that blue LFsstrongly depend on environment and time.

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9 D I S C U S S I O N

9.1 Possible peculiarity of CL0016

Galaxies in CL0016 show somewhat exceptional properties com-pared with other samples. For example, as shown in Fig. 12, thefield environment in CL0016 has a strange red LF, and the groupenvironment has an increasing faint end slope for red galaxies. Onemay find that these LFs look different compared with other LFs.The fraction of red galaxies is the highest in all environments as wesee later.

There is a concern that this peculiarity may arise from errors inthe statistical subtraction of contamination. We therefore check howthe results for CL0016 can change by changing our choice of fieldregions used in the subtraction. We take either of the two field regionsindicated by rectangles in Fig. 6 separately, rather than combiningthese two fields. We then compare the resulting CMDs and LFs forCL0016 after the statistical subtraction. This effectively correspondsto changing the field density by a factor of ∼2. Yet, we find noevidence that the results change strongly. Therefore, we suggest thatthe peculiarity of the CL0016 cluster is intrinsic. CL0016 is probablyone of the oldest systems in the Universe where red faint galaxiesare already abundant. Indeed, Butcher & Oemler (1984) illustratedthe unusually low blue fraction of this cluster for its redshift. Ourenvironments are defined on the basis of local and global (2 Mpc)density. It might be the case that even larger-scale environments (e.g.∼10 Mpc) can affect the galaxy properties (Balogh et al. 2004a; butsee also Blanton et al. 2004). In the following, we regard CL0016as an exceptional sample.

9.2 Break density

The break density corresponds to the outskirts of galaxy clusters (oneto two Rvir) and isolated groups. This means that groups and clustersare dominated by red galaxies and supports the idea that cluster-specific mechanisms have not played major roles in transforminggalaxy properties (Kodama et al. 2001a; Balogh et al. 2004a,b; DePropris et al. 2004; Tanaka et al. 2004, but see also Fujita 2004; Fujita& Goto 2004). Rather, mechanisms such as low-velocity galaxy–galaxy interactions (e.g. Mihos & Hernquist 1996) and strangulation(Larson et al. 1980) remain as strong candidates of the driver of theenvironmental dependence.

The fact that faint galaxies (MV > M∗V + 1) show a clear break

suggests that physical mechanisms actually work on faint galaxies inhigh-density environments. Then, why do not bright galaxies (MV <

M∗V + 1) show any prominent break? Of course, bright galaxies are

generally red and a break in their colours, if any, would be difficultto see. But, at least in the SDSS plot in Fig. 2, this is not the case.No break is seen even in the 25th percentile of bright galaxies. Thisimplies that the evolutionary path of bright galaxies is differentfrom that of faint galaxies in such a way that the evolution of faintgalaxies is strongly related to groups and clusters, while that ofbright galaxies is not strongly related (Tanaka et al. 2004).

Environmental dependence of galaxy properties is determinedby a priori effects (initial conditions) and a posteriori effects (en-vironmental effects). Recent near-IR studies such as the K20 Sur-vey (Cimatti et al. 2002a,b,c; Daddi et al. 2002; Fontana 2004),FIRES (Franx et al. 2003; van Dokkum et al. 2003; Rudnick et al.2003; Schreiber et al. 2004) and part of GOODS (Giavalisco et al.2004; Daddi et al. 2004; Moustakas et al. 2004; Somerville et al.2004) revealed the existence of massive red galaxies at z > 1. Al-though a significant fraction of such red galaxies are dusty starbursts

(e.g. Miyazaki et al. 2003), massive non-star-forming galaxies,which are presumably precursors of present-day ellipticals, do exist.Since their colours match with passive evolution, their properties areexpected to be largely determined by a priori effects and subsequentenvironmental effects are not very important. That is, their star for-mation rate is already low before environmental mechanisms playa role. These massive galaxies would evolve to bright galaxies inour definition, and would explain the trends observed in Section 5.On the other hand, the deficit of red faint galaxies is a function ofenvironment, in a way that red faint galaxies preferentially populatein high-density environments. Therefore, we suggest that propertiesof bright galaxies are largely determined by a priori effects, whilethose of faint galaxies are largely determined by a posteriori effects.

9.3 The build-up of the CMR

In Section 7, we observed the build-up of the CMR. To further quan-tify the build-up, we base our analysis on stellar masses of galaxies.If galaxies stop their star formation, they will be fainter (�MV ∼ 1)while keeping their stellar masses nearly unchanged. Since our stel-lar mass estimates are not very accurate, we cannot examine thedetailed shape of the stellar mass function. Instead, we investi-gate the giant-to-dwarf number ratio (g/d). Giants and dwarvesare defined as those having log10(M ∗/M�) > 10.6 and 9.7 <

log10(M ∗/M�) < 10.6, respectively. Note that log10(M ∗/M�) =10.6 corresponds to MV = −20.5, −20.3, and −19.8 for red galaxiesat z = 0.83, 0.55, and 0, respectively.

Since our estimates of stellar masses are primarily based on therest-frame V-band magnitudes, they are affected by recent/on-goingstar formation activities. In fact, as shown by the iso-mass curve inFig. 8, the V-band magnitude can change by 1.5 mag for the samestellar mass between passively evolving galaxies and the constantlystar-forming galaxies. This is translated to the variation in mass-to-light ratio by a factor of ∼4. This number can be viewed as a solidupper limit of the uncertainties in M∗ in a relative sense, since wecorrect for such variation in mass-to-light ratio by applying SED fit-ting when deriving stellar masses (note that absolute stellar massesdepend on other factors such as the stellar initial mass function).Moreover, since we mainly discuss the red galaxies, the actual vari-ation in mass-to-light ratio must be much smaller (less than a factorof 2). In what follows, we focus on the red galaxies. We separatelydiscuss field, group and cluster environments.

Field environment: Fig. 13 shows g/d in three different envi-ronments and at three different redshifts. The parameter g/d clearlydepends on both environment and time. A general trend is that g/dis largest in the field environment at any redshifts, suggesting thatred faint galaxies are relatively rare in the field. This is consistentwith our finding in Fig. 8 that the faint end of the field CMR is notclear. The field g/d ratio is the largest in SDSS. This is likely to bedriven by the build-up of the bright end of the field CMR (i.e. thefraction of giants increases).

Group environment: The g/d ratio in groups is the largest inRX J0153 and it decreases at low redshifts. This decrease in g/dmay reflect the build-up of the group CMR at the faint end: thefraction of faint red galaxies increases.

Cluster environment: The cluster g/d also shows the decreaseat low redshifts. Although we could not see a prominent build-upof the cluster CMR in Section 7, the g/d evolution may suggestthe faint end of the cluster CMR is still under construction evenat z = 0.83.

It seems that the bright end of the CMR appears first, and thefaint end is filled up later on. A likely scenario for this build-up is

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Figure 13. The giant-to-dwarf ratio of red galaxies plotted against envi-ronments. The meanings of the lines are shown in the figure. The errors arebased on Poisson statistics.

that blue galaxies stop their star formation and fade to settle downon to the CMR, and the truncation of star formation starts frombright (massive) galaxies. This scenario involves suppression of starformation activity in blue galaxies. How do blue galaxies in groupsand field environments fade and settle exactly on the same CMR asthat of cluster galaxies? We recall that we found no evidence forenvironmental dependence of the slope of the CMR. If the CMRis a product of the mass–metallicity relation (Kodama & Arimoto1997), then blue galaxies have to ‘know’ the metallicities of redgalaxies they settle on to. Since blue galaxies in the field and groupenvironments should follow quite different star formation historiesfrom those of red galaxies in clusters, one may expect that theirmetallicities are quite different. This is not the case, however. Animportant point here is that we are considering relatively massive(log10 M ∗ > 9.7) galaxies. The cosmic star formation rate declinesat z < 1 (e.g. Madau et al. 1996; Madau, Pozzetti & Dickinson1998; Fujita et al. 2003), and therefore their major episode of starformation took place at higher redshifts. Subsequent star formationdoes not strongly enrich metals in galaxies, and metals locked in theirstars do not grow significantly after major star formation (Tinsley1980). Therefore, once galaxies mature, the epoch when they stoptheir star formation is not a major concern from a chemical point ofview. Whenever they stop their star formation, they will settle on tothe CMR.

To quantify the build-up of the CMR further, we plot in Fig. 14the fraction of red galaxies as functions of environment, stellar massand time. As described in Section 3.3, we cannot deny the possibilitythat we miss a fraction of blue galaxies in RX J0153 and CL0016.Therefore, the real red fractions in RX J0153 and CL0016 would besmaller than those shown in the plot. We discuss the red fraction inthe field, group and cluster environments separately as follows.

Field environment: The massive galaxies show an increase inthe red fraction, which is consistent with the build-up of the brightend of the field CMR. On the other hand, the least massive galaxiesshow a decrease. However, this may be because we tend to miss afraction of blue galaxies in RX J0153 and CL0016 due to our photo-zselection.

Group environment: Due to the relatively large errors, the redfraction of group galaxies is consistent with being unchanged withredshift.

Cluster environment: Within the errors, the red fraction is con-sistent with being almost constant over the time under study. This

.

Figure 14. Fraction of red galaxies plotted against the look-back time (thecorresponding redshift is shown in the top ticks) for the three stellar massbins. The meanings of the lines are shown in the figure. Each point is shiftedhorizontally by a small amount for clarity. The error bars show the Poissonerrors.

is in agreement with the trend that no significant CMR build-up isseen in clusters within the redshift range explored. There is a hint ofevolution, however, at the least massive bin, where the red fractionmay increase from z = 0.83 to lower redshifts.

The overall trend is that the red fraction is lowest in the fieldenvironment and higher in groups and clusters. But, in any environ-ments and at any redshifts, the red fraction is higher for more massivegalaxies. That is, the red fraction is higher in a denser environmentand for more massive galaxies.

9.4 Implications for galaxy evolution

From Figs 13 and 14, we consider that the evolutionary stage of theCMR build-up is different in different environments. The bright endof the cluster CMR is already built by z = 0.83, and only the faintend shows a possible evolution at z < 0.83. On the contrary, in thefield regions, the bright end of the CMR is being vigorously built,while the build-up at the faint end has not yet started.

A possible interpretation of these results is that strong evolutionhas occurred since z ∼ 1 in a ‘down-sizing’ way. It was Cowie et al.(1996) who first pointed out the down-sizing galaxy evolution. Athigh redshifts, massive galaxies actively form stars. At lower red-shifts, massive galaxies show less active star formation and the mainpopulation of active star formation moves to less massive galaxies.The bright end of the CMR is formed first as massive galaxies stoptheir star formation, and the build-up proceeds to the faint end asless massive galaxies stop their star formation. Our results suggestthat the evolutionary stage of this down-sizing depends on environ-ment. The evolution of massive cluster galaxies is almost completedby z = 0.83. In the field environment, the evolution of massivegalaxies is strong, while that of less massive galaxies is found tobe weak. Therefore, it seems that the main population that showsstrong evolution is shifted to higher mass galaxies in lower densityenvironments.

One of the drivers of the environmental dependence of the down-sizing could be initial conditions of galaxy formation (a priori ef-fects). Galaxies are formed earlier in higher density peaks of theinitial density fluctuation of the Universe. Thus, galaxies in clusters

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are naturally at an advanced stage of galaxy evolution comparedwith those in lower density environments. This may explain the en-vironmental dependence of the build-up of the CMR. We consider,however, that the environmental dependence of the down-sizing ef-fect is not solely caused by the a priori effects, but environmental(or a posteriori) effects should contribute significantly. Effects thatsuppress star formation activities are strong in high-density envi-ronments (Section 5), and they accelerate the build-up of the CMR.

We observed the build-up of the CMR at the faint end in groupenvironments. This should mean either that faint galaxies are stillactively forming stars, or that less massive galaxies are not yet fullyformed. We cannot, however, discriminate these possibilities withthe data in hand. To do this, we need to estimate stellar masses offaint blue galaxies accurately. Further discussion on the driver ofthe CMR build-up awaits more accurate stellar mass estimates bynear-infrared data.

Detailed observations of early-type galaxies suggest that the typ-ical luminosity-weighted age of field early-type galaxies is youngerthan cluster galaxies (e.g. Kuntschner et al. 2002; Gebhardt et al.2003). Thomas et al. (2005) examined nearby early-type galaxiesand suggested that formation of early-type galaxies is earliest inhigh-density environments and delayed by ∼2 Gyr in low-densityenvironments, and the formation of massive galaxies predates thatof less massive galaxies. Utilizing near-IR data, Feulner (2005) andJuneau et al. (2005) reported that the epoch of major star formationtook place at higher redshifts for more massive galaxies, and starformation is more extended in time for less massive galaxies. Allthese studies lend support to the down-sizing picture.

We close our discussion by noting some caveats on our results.Our results are based only on two high-z clusters, which may notbe a typical cluster at each redshift (particularly CL0016). To reacha firm conclusion, we need to observe more clusters at various red-shifts. Also, errors in the photometric redshifts and stellar massestimates remain as a concern. Near-infrared (e.g. K-band) data arerequired to improve them. Spectroscopic data are clearly needed tofurther address the effectiveness of the photometric redshifts andassess errors in the statistical contamination subtraction. We hopeto overcome these uncertainties in our future paper.

1 0 S U M M A RY A N D C O N C L U S I O N S

We began this paper by introducing the three axes on which galaxyproperties strongly depend, namely, environment, stellar mass andtime. We found that the star formation activity in galaxies is indeeddependent on all of these three axes and galaxies follow complicatedevolutionary paths.

We conducted multiband imaging of two high-z clusters,RX J0153 at z = 0.83 and CL0016 at z = 0.55, with Suprime-Cam on Subaru. These Subaru data were combined with the SDSSdata (z = 0), and we carried out statistical analyses of galaxy prop-erties as functions of environment, mass and time. We examined thecolour–density relations, colour–magnitude diagrams (CMDs), andluminosity functions (LFs) of galaxies.

First, we applied our photometric redshift technique to RX J0153and CL0016 fields to largely eliminate fore-/background contami-nation and discovered large-scale structures surrounding the mainclusters. Details are described in Kodama et al. (2005).

Then we examined the relationship between galaxy colours andenvironments. It was found that galaxy colours abruptly change atthe break density at any redshifts considered in this paper. Faintgalaxies (MV > M∗

V + 1) show a prominent break, while brightgalaxies (MV < M∗

V + 1) do not show such a strong break. Based

on the break density, we defined three environments: field, groupand cluster.

We examined CMDs in the three environments. The highlight ofthe CMD analysis was to show how the CMR is built up as functionsof time, environment and mass. There is no clear CMR in the fieldregions in RX J0153, while a clear CMR can be seen in the sameenvironment in CL0016 and SDSS, particularly at the bright end. Ingroups of RX J0153, a clear CMR is visible only at the bright end(MV < −20). The relation is found to extend down to our magni-tude limit in groups of CL0016 and SDSS. Clusters have clear CMRsdown to our magnitude limits at all redshifts considered here, with apossible remaining activity at the faint end. These trends are quan-tified by the scatter of the CMRs.

The build-up of the CMR was quantified by the giant-to-dwarfratio (g/d) and the red fraction. From z = 0.83 to z = 0, the field g/dincreases and the group g/d decreases, suggesting the CMR build-up at the bright end in the field and at the faint end in the group.The cluster g/d also decreases. This may imply that the faint end ofthe cluster CMR was still under construction at high redshifts. Wefound that the red fraction is higher in denser environment and formore massive galaxies.

As a possible interpretation of these results, we suggested thatgalaxies evolve in a down-sizing way. That is, the main populationsthat host active star formation are shifted from massive galaxiesto less massive galaxies with time. It is likely that an evolutionarystage of the down-sizing depends on environment. Cluster galaxiesevolve most rapidly, and group and field galaxies follow. All inall, it seems that galaxy evolution proceeds from massive galaxiesto less massive galaxies, and from high-density environments tolow-density environments. In order to confirm this trend, however,further studies are obviously needed. We need to increase the clustersample and also to perform follow-up observations such as near-infrared imaging and spectroscopic surveys.

AC K N OW L E D G M E N T S

MT acknowledges support from the Japan Society for the Promo-tion of Science (JSPS) through JSPS research fellowships for YoungScientists. We thank Fumiaki Nakata and Masafumi Yagi for theirhelp during the data reduction. We thank the anonymous referee forthe careful reading of the paper and invaluable comments which im-proved the paper significantly. This work was financially supportedin part by a Grant-in-Aid for the Scientific Research (No. 15740126,16540223) by the Japanese Ministry of Education, Culture, Sportsand Science. This study is based on data collected at Subaru Tele-scope, which is operated by the National Astronomical Observatoryof Japan.

Funding for the creation and distribution of the SDSS Archivehas been provided by the Alfred P. Sloan Foundation, the Partic-ipating Institutions, the National Aeronautics and Space Admin-istration, the National Science Foundation, the US Department ofEnergy, the Japanese Monbukagakusho, and the Max Planck So-ciety. The SDSS Web site is http://www.sdss.org/. The SDSS ismanaged by the Astrophysical Research Consortium (ARC) forthe Participating Institutions. The Participating Institutions are TheUniversity of Chicago, Fermilab, the Institute for Advanced Study,the Japan Participation Group, The Johns Hopkins University, theKorean Scientist Group, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institutefor Astrophysics (MPA), New Mexico State University, Univer-sity of Pittsburgh, Princeton University, the United States NavalObservatory, and the University of Washington.

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A P P E N D I X A : D E TA I L S O F T H E VO L U M EC O R R E C T I O N

In a flux-limited sample, intrinsically faint galaxies are observedonly at low redshifts, while intrinsically bright galaxies can be ob-served at higher redshifts. For example, in our SDSS sample, wecan observe galaxies brighter than M V, vega < −19.5 at all redshifts,while M V, vega = −18 can be observed only at z < 0.033. We there-fore need to correct for this incompleteness when we discuss galaxyproperties. A simple way to do this is to give heavier statisticalweights to intrinsically fainter galaxies according to their redshifts.A flux-limited sample can mimic a volume-limited one in this way.A statistical weight can be computed as

wi = Vsurvey/Vi,max, (A1)

where wi is the statistical weight of i-th galaxy, V survey is the volumecontained within our redshift range (0.005 < z < 0.065), and Vi,max isthe maximum volume over which the i-th galaxy could be observed(e.g. in case of MV = −18.0, V max is the volume contained within0.005 < z < 0.033). We assume that an LF does not evolve withinthe surveyed redshift range.

We can test the effectiveness of this volume correction by sim-ulating the galaxy distribution with a given LF. For simplicity, weadopt a flat LF such that galaxies are uniformly distributed within amagnitude range of −21 < M < −15. We examine a set of galaxydistribution here.

In the top panels of Fig. A1, we show a model of uniform galaxydistribution in a given volume, and the middle of the top panelsshows a volume corrected LF. As seen in the panel, an LF is recon-structed well. In the middle panels, we set the non-uniform galaxydistribution in a given volume. If galaxies are uniformly distributedin a given volume, we expect that the number of galaxies in a given

Figure A1. The volume-corrected LFs of three types of galaxy distribu-tion. From top to bottom, the panels show a uniform distribution (N gal ∝ r2),an r-independent distribution (N gal is constant at a given r), and a uniformdistribution with redshift gaps. The left-hand panels show the galaxy distri-bution as a function of a distance from us (units are arbitrary). The middlepanels present reconstructed LFs based on the wi weighting. The right-handpanels show reconstructed LFs based on the w i,corr weighting. See text fordetails.

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radial shell scales as N gal ∝ r 2 (this is the case for the top panels).But, here we adopt the distribution such that N gal is independent ofr. In this case, galaxies are not distributed uniformly in a volume,and a volume-corrected LF does not reconstruct the parent LF. Thisis expected – the volume correction assumes the uniform galaxydistribution over a survey volume. The bottom panels show anotherexample; the radial galaxy distribution has ‘gaps’. The volume cor-rection again fails to reproduce the parent LF.

Therefore, we take into account variations in the radial galaxydistribution such as large-scale structures when we perform the vol-ume correction. In our sample, galaxies having MV < −19.5 can beobserved regardless of redshift (i.e. volume-limited). We use thesegalaxies to correct for the radial variation. A number of galaxieswith MV < −19.5 should scale as ∝ r2 if they are distributed uni-formly in the Universe. We define the radial distribution of galaxieshaving MV < −19.5 divided by r 2 as f (r ), and radially averagedf (r ) as f ave. The statistical weight becomes:

wi,corr = (Vsurvey/Vmax) × [ fave/ f (r )] . (A2)

Note that, in the first term, we do not include the volume where nogalaxies are observed. For example, we do not calculate the vol-ume contained in the ‘gaps’ in the bottom panels in Fig. A1. Inthe latter term, we make radial bins to compute f (r ) and f ave. Weconfirmed that our conclusions are not strongly affected by uncer-tainties arising from the binning. The right-hand panels of Fig. A1shows volume-corrected LFs using this equation. The parent LFsare reconstructed well. Note that equation (A2) does not reproducethe absolute number of galaxies.

As a final check of this correction, we use a real galaxy distributionin our sample and artificially assign their magnitudes based on a flatLF. We confirm that the parent LF is recovered well in all the field,group and cluster samples by using equation (A2). It turns out thatthe correction for the non-uniformity is particularly important forgroup and cluster galaxies whose distribution is strongly related tolarge-scale structures. Actually, cluster galaxies have ‘gaps’ in theirradial distribution. To sum up, a flux-limited sample can be usedto discuss galaxy properties by applying the statistical weights ofequation (A2).

A P P E N D I X B : S TAT I S T I C A LC O N TA M I NAT I O N S U B T R AC T I O N

This appendix describes the statistical contamination subtractionprocedure in detail. See Fig. B1 for help. We use the control fieldsample defined in Section 3 to subtract the contamination. For sim-plicity, in this appendix, we refer to the control field sample as thefield sample. The field sample examined in the main text should beconsidered as the ‘target’ sample here.

Let us pick a galaxy in a target CMD. We move on to a CMD offield galaxies, and pin down a point where our target galaxy shouldlie (i.e. the magnitude and colour of the galaxy we picked). Then wedraw a circle (strictly speaking, an ellipse) around the point on thefield CMD, and count field galaxies that fall in the circle. The aboveprocedure is performed for all target galaxies, and then we knowhow many times a field galaxy is counted by the target galaxies.We define the statistical weight for a field galaxy as the inverse ofits count. For example, if a field galaxy is counted three times (i.e.three target galaxies share the field galaxy), the statistical weight ofthe field galaxy is 1/3. Then we define the probability for a targetgalaxy to be a field galaxy as the ratio of the sum of statisticalweights of field galaxies in the circle of the target galaxy to thetotal number of field galaxies. We randomly pick up a target galaxy,

Figure B1. The contamination subtraction procedure.

throw a dice, and determine if the target galaxy is a field galaxy ornot according to its field probability. We repeat this procedure untilwe subtract sufficient number of galaxies. In short, this procedurecan be considered as a generalized form of the grid-based method(Kodama & Bower 2001b; Pimbblet et al. 2002). It should be notedthat since we make no grids, we are free from uncertainty on howwe make the grids. Also note that we are free from the ‘negativegalaxy’ problem which the grid-based method faces.

There are two parameters in our method, namely, the radius andthe shape of the aperture within which we count field galaxies. Theaperture we actually use is shown in Fig. B1. It should be noted thatour results do not depend on the size and shape of the aperture inreasonable ranges.

How many galaxies should we subtract? When we have galaxiesthat occupy a surface area of A arcmin2, we expect the contam-ination to be �fieldA galaxies, where �field is the average surfacedensity of field galaxies. Since we define the environment by thegalaxy density, it is not very straightforward to estimate a surfacearea occupied by density-selected galaxies. We infer a surface areaby a simple approximation. Because the inverse of a galaxy densitymeans the average surface area occupied by a galaxy (i.e. unit ofarcmin2 galaxies−1), the sum of the inverse of the density gives thesurface area:

A� =N∑

i=1

1

�local,i, (B1)

where � local is the local density and i runs through the first to theN-th galaxy we select. We show in Appendix C that this estimateis statistically robust. A Poisson error is then added in �fieldA� ineach Monte Carlo realization of the contamination subtraction.

We check if our statistical contamination subtraction techniqueproperly subtracts field galaxies. We examine if the subtracted tar-get galaxies reproduce the CMD of field galaxies. Fig. B2 plotsthe distribution of subtracted target galaxies on a CMD projectedon to one-dimensional space for clarity. It can be seen that the

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Figure B2. Field galaxies in RX J0153 are divided into six mag bins (sixpanels from top to bottom), and fractions of field galaxies as a functionof R − z′ colour are plotted. The open squares show the distribution offield galaxies. The solid circles show the distribution of ‘subtracted’ targetgalaxies.

distribution of subtracted galaxies closely follows that of field galax-ies. The very small difference between them is due to the fact that weapply a smoothing of the galaxy distribution on the field CMD whenwe estimate field probabilities (the size and shape of the aperturedetermines the smoothing scale). Despite the small difference wesee a fairly good correspondence, and we therefore conclude thatour subtraction method properly removes the field contamination.

A P P E N D I X C : C O N V E R S I O N F RO M D E N S I T YTO S U R FAC E A R E A

In Appendix B, we converted from local density to a surface area thatis occupied by density-selected galaxies. We examine the validityof this approximation in this section. We construct a simple toymodel of a galaxy cluster and investigate differences between thereal (metric) area and the area estimated from the density.

Our toy model consists of uniform field galaxies and cluster galax-ies whose distribution follows the King model (King 1962). We ran-domly generate galaxies and estimate the density for each galaxy bythe nearest-neighbour method used in this paper. Galaxies within afixed aperture around the cluster are selected. Then we compare thearea of the aperture (Aaper) with the area estimated from densitiesof galaxies within the aperture (A� , see equation B1). We calculate

Figure C1. A�/Aaper plotted against the number ratio of cluster galaxies tofield galaxies in the aperture. The points mean the median of the distribution,and the error bars represent the quartiles of the distribution.

A�/Aaper in each realization and repeat 1000 times. There are twoparameters in the model: the radius of the aperture, and the frac-tion of cluster galaxies to field galaxies in the aperture. The latterparameter determines the richness of a cluster.

As a fiducial value, the aperture radius is set to 3rc (rc meansthe core radius of the King model). Fig. C1 plots A�/Aaper againstthe number ratio of cluster galaxies to field galaxies in the aperture.The result is encouraging – A�/Aaper is almost unity. In the densitycalculation, the distance to the nth nearest galaxy is used (n = 10 isadopted in the main text). We run the simulation using n = 5, 10, 20,and find that there is a weak trend that the scatter in A�/Aaper

decreases with increasing n, but the median is always close to unity.Our choice of the aperture radius seems to have a rather strong effect.The scatter in A�/Aaper increases as we adopt a smaller aperture(e.g. 2rc). On the other hand, if we adopt a larger aperture, say 5r c,the scatter reduces to almost half. However, in any case, the medianA�/Aaper is always close to unity. Therefore, we conclude that A�

is a good measure in a statistical sense.

A P P E N D I X D : G L O BA L D E N S I T Y

Our aim in this appendix is to address how effective and quantitativeour group/cluster separations are. For simplicity, groups and clustersare referred to as systems in this appendix.

We run the friends-of-friends algorithm (FOFA; Huchra & Geller1982) in our SDSS sample to find galaxy groups and clusters. TheFOFA is a famous group/cluster finding algorithm and its statisticalproperties are well known (Moore, Frenk & White 1993; Frederic1995; Ramella, Pisani & Geller 1997; Diaferio et al. 1999; Merchan& Zandivarez 2002). The basic algorithm of the FOFA is to find a setof galaxies connected within certain linking lengths to each other.There are two linking lengths, namely, an angular separation (D0)and a line-of-sight velocity difference (V 0). Since galaxy systemsare elongated along the line of sight in redshift space (‘Finger ofGod’ effect), we need to handle D0 and V 0 separately.

We run the FOFA in a volume-limited sample defined by 0.010 <

z < 0.065 and MV < −19.5 so that we do not have to apply anyscaling in the linking lengths. The FOFA parameters are set toD0 = 700 kpc and V 0 = 400 km s−1. These values are slightlydifferent from those adopted in Tanaka et al. (2004). Our conclu-sions are not, however, strongly dependent on the choice of the

Figure D1. Global density plotted against N gal determined by the FOFA.The dot and the error bars show the median and quartiles of the global densitydistribution in each system.

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parameters. There is one more parameter, N min, which determinesthe minimum size of galaxy systems. If a system has fewer membersthan N min, the system is not considered here. We set N min = 5 as inTanaka et al. (2004) since a significant fraction of N < 5 groups areexpected to be spurious (Frederic 1995; Ramella et al. 1997, 2002).

In Fig. D1 we present a correlation between the number of mem-ber galaxies (N gal) of the FOFA systems and the global density ofthe member galaxies. The correlation is encouragingly good – thereis a positive correlation between the two quantities, especially atN gal > 20. N gal < 20 systems show only a weak correlation. This isprobably because the typical extent of such poor systems is smallerthan 2 Mpc and the number fluctuation of field galaxies contributesto the global density. As for N gal > 20, cluster members dominatethe 2 Mpc aperture and hence we have a good correlation there. Afew systems that have small N gal with high global density can be

seen. They are either parts of rich systems accidentally split by theFOFA or real isolated systems close to nearby rich systems. Theycomprise only a small fraction and the effect of such contaminationis negligible. In summary, the global density is a powerful tool toseparate groups from clusters quantitatively.

We measure velocity dispersions of detected systems using thebiweight estimator (Beers, Flynn & Gebhardt 1990). By examiningthe relationship between N gal and the velocity dispersion of thesystem (σ ), we find that the global density threshold used in the maintext for the SDSS, �global = 2.7, roughly corresponds to systemshaving σ= 300–400 km s−1. Therefore, our groups typically haveσ< 300–400 km s−1 and clusters have σ > 300–400 km s−1.

This paper has been typeset from a TEX/LATEX file prepared by the author.

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